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Poincaré’s discovery of homoclinic points. (English) Zbl 0812.01011

The author claims that the most radical break with prevailing conceptions was Poincaré’s discovery of homoclinic points, which nowadays figure in studies of “chaotic” motions. The presence of a homoclinic point in a dynamical system complicates the orbit structure considerably and implies the existence of trajectories with quite unpredictable behaviour. Poincaré first encountered homoclinic points in 1889 in connection with his memoir for which he had been awarded a prize by the Swedish king Oscar II. The purpose of this article is to throw some light on the events connected with the memoir. The author is citing documents of the Mittag-Leffler Institute from which one can obtain a clear picture of what actually happened.

MSC:

01A55 History of mathematics in the 19th century
70F07 Three-body problems
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References:

[1] Birkhoff, G. D. 1935. Nouvelles recherches sur les systèmes dynamiques, Mem. Pont. Acad. Novi Lyncaei 1, 85-216. · JFM 60.1340.02
[2] Mittag-Leffler, G. 1912. Zur Biographie von Weierstrass, Acta Math. 35, 29-65. · JFM 42.0017.04
[3] Poincaré, H. 1881-1886. Mémoire sur les courbes définies par une équation différentielle I?IV, Journal de Mathématiques pures et appliquées, 3e série, t. VII, 375-422, t. VIII, 251-296; 4e série, t. I, 167-244, t. II, 151-217.
[4] Poincaré, H. 1890. Sur le problème des trois corps et les équations de la dynamique, Acta Math. 13, 1-271.
[5] Poincaré, H. 1892-1899. Les méthodes nouvelles de la mécanique céleste I?III, Gauthiers-Villars.
[6] Poincaré, H. 1921. Lettres à M. Mittag-Leffler concernant le mémoire couronné du prix de S.M. le roi Oscar II, Acta Math. 38, 161-173.
[7] Smale, S. 1965. Diffeomorphisms with many periodic points, in Differential and Combinatorial Topology, Princeton University Press, 63-80. · Zbl 0142.41103
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